Arnesen moved from childhood club Fremad Amager, in the lower leagues of Danish football, to the big Dutch team Ajax in November/December 1975. At the time he was only 19, and he made the move together with Fremad Amager teammate Søren Lerby, who was only 17. Arnesen got his debut with Ajax 3 or 4 months later, on 7 March 1976, in a 1–1 draw with FC Utrecht. It was also in his time at Ajax, that Arnesen debuted for the Danish national team, in a 1977 friendly 1–0 loss against Sweden, in Malmö. In Arnesen's six seasons at Ajax, he won three Dutch Eredivisie championship titles in 1977, 1979, and 1980, and won the KNVB Cup in 1979, beating FC Twente 3–0 in the final. In 1980, Arnesen and Ajax reached the semi-final of the European Cup, before they were eliminated by English team Nottingham Forest (2–0 away loss, 1–0 home win).

In the summer of 1981, Arnesen was bought by Spanish side Valencia CF, and two years later he went to Belgium to play for RSC Anderlecht. In November 1985, Arnesen returned to Holland to play for Ajax' rivals PSV Eindhoven. Here he saw three extremely successful seasons, winning the Eredivisie three years in a row, in 1986, 1987, and 1988, with fellow Danes Jan Heintze, Søren Lerby and Ivan Nielsen also in the team. Arnesen stopped his international career in 1987, after the Danish team qualified for the Euro 1988. In 1988, he won the KNVB Cup once more, and he was part of the PSV team that conquered the European Cup, though he did not play in the final match due to injury.